It's extremely fast, yes, and theoretically could be made compatible with a TCP/IP network (devices have been made on earth to use laser transmission as a long distance bridge between networks) but you've still got a ping time measured in hours. The LCRD does not bypass the speed of light limitation. You'd be able to get on the internet as long as you queue up web pages long enough ahead of time (an earth based proxy would be required as timeouts would prevent a direct bridge from working), but real time games would be beyond impossible.
From the moon, a connection wouldn't be stable, you'd be taking a typical ping time and adding ~1300 ms to it - even assuming the earth and moon based network latency is somehow 0 that's not going to be playable, though you may be able to stay connected enough to do doffs, reputation, or fleet projects.
Well, I guess they could do it for the setting? Eight years ago I played the Saladin campaign in Age of Empires II on my laptop while sitting in the Middle Eastern exhibit wing of the Metropolitan Museum of Art.
Member since November 2009... I think.
(UFP) Ragnarök Fleet - Logistics Division Vice Admiral T'Phira; Captain Selena; Captain Altecha; Captain K'Kera; Lieutenant Aydihe Tefx; Subcommander S'ena
(KDF) Lieutenant General Vokno (aka Lady Vokno); Lieutenant General Raylene